Nothing quite gets the blood racing like this topic. So I thought I would briefly write about some aspects of the question.
Firstly, of course, it must be appreciated that there is no such thing as an amnesty officially acknowledged. Such material as there is inhabits the blandly designated territory amongst the Home Office policies "Enforcement Instructions", under Chapter 53, "Extenuating circumstances". In their introduction these inform us that the Secretary of State will not seek to remove people incompatibly with their human rights: but will also forgo removal where there are "compelling reasons, usually of a compassionate nature, for not doing so in an individual case."
Aside from this general discretion, special attention is given to cases where there has been a delay in considering the case where fault lies at the door of the Home Office. This was discussed by the House of Lords in HB Ethiopia at -. But the gloss the Home Office put on the case is very interesting. They consider relevant delay to kick in where these three factors coincide: (a) two years passing without a decision on an outstanding application; where (b) the individual has made progress enquiries; and that in the meantime (c) they have either (i) built up significant private or family life, or (ii) would suffer considerable hardship if expulsion proceeded. These factors are then developed with some examples involving longer time periods and various family links (or, more interestingly, the subsequent text may be read as comprising free-standing categories of benevolence: in which case any timely application for further leave to remain not answered in 3-5 years creates a presumption of a grant of leave absent bad character). These are generous criteria compared to the approach that many Immigration Judges have taken in recent years.
Perhaps the days of conservatism are over, though, with the new personnel in the higher immigration judiciary. One of the more famous government concessions in recent years was the Seven Year Child Concession, the best location for which is the decision of Rix LJ in the Court of Appeal in NF (Ghana) v Secretary of State for the Home Department  EWCA Civ 906 (30 July 2008) . New Upper Tribunal Immigration President Blake J has pointed out that the withdrawal of that policy does not destroy its underlying rationale, for as he puts it, the historic identification of seven years of residence of a child as one that would presumptively require regularisation of immigration status of child and parents in the absence of compelling countervailing factors was really an administrative way of giving effect to the principle of the welfare of the child as a primary consideration, as required by the UN Convention on the Rights of the Child 1989 which stresses the best interests of the child. See LD (Article 8 best interests of child) Zimbabwe  UKUT 278 (IAC) (10 August 2010).
So, whatever the official government spin, there are de facto amnesties in law and policy.